We have investigated dielectric losses in amorphous SiO thin films under operating conditions of superconducting qubits (mK temperatures and low microwave powers). For this purpose, we have developed a broadband measurement setup employing multiplexed lumped element resonators using a broadband power combiner and a low-noise amplifier. The measured temperature and power dependences of the dielectric losses are in good agreement with those predicted for atomic two-level tunneling systems (TLS). By measuring the losses at different frequencies, we found that the TLS density of states is energy dependent. This had not been seen previously in loss measurements. These results contribute to a better understanding of decoherence effects in superconducting qubits and suggest a possibility to minimize TLS-related decoherence by reducing the qubit operation frequency.PACS numbers: 85.25.Cp, 77.22.Gm, 74.50.+r, 73.61.Jc For the realization of a quantum information processor based on superconducting circuits, physical mechanisms responsible for decoherence have to be cleared up 1 . Microwave resonators are key ingredients of superconducting quantum circuits. Dielectric thin films are widely used in qubit fabrication as insulating layers and overlay capacitors of the resonators. Lumped capacitors are often employed in qubits to shunt Josephson junctions in order to adjust their operation frequency. It is well established 2 that two-level tunneling systems (TLS) in insulating films contribute to dielectric losses at microwave frequencies and thus often appear as dominant decoherence sources, in particular for superconducting phase qubits. In this paper, we experimentally study microwave properties of amorphous dielectric thin films embedded in superconducting lumped element resonators.Dielectric losses induced by TLS in the qubit working regime have been extensively studied in recent years and good agreement with the predictions of the TLS tunneling model has been found 2-6 . However, most of the investigations were done using coplanar waveguide resonators, where TLS and thus losses originate from surfaces 3-6 . The bulk losses in dielectric films may also play a significant role for a variety of qubit designs and till now remain less understood. Furthermore, there is little known about the energy dependence of the TLS density of states n(E) in typical qubit materials. A better understanding of this matter would allow to improve qubit coherence and lead to a better qubit design and operation. In the standard tunneling model (TM) of glasses, a dependence of n(E) on energy is not excluded 7 , but characteristic glassy behavior refers to an essentially constant density of states 8,9 . So far, a slight dependence of n(E) on energy had been extracted from specific heat measurements 10 , but could be explained in terms of the TM taking into account the time scales of the measurements 7 . Earlier investigations of the sound velocity in glassy thin a-SiO x films in the kHz 11 and 300 MHz 12 ranges validate the constancy of the den...